Constructing Hopf Insulator from Geometric Perspective of Hopf Invariant

We propose a method to construct Hopf insulators based on the study of topological defects from the geometric perspective of Hopf invariant I. Firstly, we prove two types of topological defects naturally inhering in the inner differential structure of the Hopf mapping. One type is the four-dimensional point defects, which lead to a topological phase transition occurring at the Dirac points. The other type is the three-dimensional merons, whose topological charges give the evaluations of I. Then, we show two ways to establish the Hopf insulator models. One approach is to modify the locations of merons, thereby the contributions of charges to I will change. The other is related to the number of defects. It is found that I will decrease if the number reduces, while increase if additional defects are added. The method developed in this study is expected to provide a new perspective for understanding the topological invariants, which opens a new door in exploring and designing novel topological materials in three dimensions.